(… for a similar continuous model).
Assume birth rate \(b\) and death rate \(d\), and growth rate \(r = b-d\).
The mean of the process is also exponential growth
\[E(N_t) = N_0 e^{r t}\] If birth rate = death rate:
\[SD(N_t) \approx \sqrt{2 N_0 b t}\]
Note, increases as square root of time. There’s a somewhat more complex formula for births \(\neq\) deaths.
More importantly:
\[ P(extinction) = \begin{cases} \text{if}\,\,b > d;& (d/b)^{N_0}\\
\text{if} \,\, d > b;& 1 \end{cases}\] Note that even when birth rate = death rate, \(P(extinction) = 1\), i.e. eventual extinction is certain. This is very similar to the eventual probability of fixation for genetic drift.
Also, probability of extinction is lower for smaller \(N_0\).